Chapter 14
Introduction to the Use of Bayesian Methods for
Reliability Data
William Q. Meeker and Luis A. Escobar
Iowa State University and Louisiana State University
14 - 1
Copyright 1998-2008 W. Q. Meeker and L. A. Escobar.
Based on the authors’ text Statistical Methods for Reliability
Data, John Wiley & Sons Inc. 1998.
December 14, 2015
8h 9min
Introduction
• Bayes methods augment likelihood with prior information.
• A probability distribution is used to describe our prior beliefs
about a parameter or set of parameters.
• Sources of prior information:
Subjective Bayes: prior information subjective.
Empirical Bayes: prior information from past data.
• Bayesian methods are closely related to likelihood methods.
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Updating Prior Information Using Bayes Theorem
Bayes Theorem provides a mechanism for combining prior
information with sample data to make inferences on model
parameters.
For a vector parameter θ the procedure is as follows:
• Prior information on θ is expressed in terms of a pdf f (θ ).
• We observe some data which for the specified model has
likelihood L(DATA|θ ) ≡ L(θ ; DATA).
R(θ )f (θ )
L(DATA|θ )f (θ )
=R
L(DATA|θ )f (θ )dθ
R(θ )f (θ )dθ
• Using Bayes Theorem, the conditional distribution of θ given
the data (also known as the posterior of θ ) is
f (θ |DATA) = R
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b ) is the relative likelihood and the
where R(θ ) = L(θ )/L(θ
multiple integral is computed over the region f (θ ) > 0.
Introduction to the Use of Bayesian Methods for
Reliability Data
Chapter 14 Objectives
• Describe the use of Bayesian statistical methods to combine
prior information with data to make inferences.
• Explain the relationship between Bayesian methods and likelihood methods used in earlier chapters.
• Discuss sources of prior information.
• Describe useful computing methods for Bayesian methods.
• Illustrate Bayesian methods for estimating reliability.
• Illustrate Bayesian methods for prediction.
• Compare Bayesian and likelihood methods under different
assumptions about prior information.
Posterior
f ( θ |DATA )
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• Explain the dangers of using wishful thinking or expectations as prior information.
Likelihood
L ( DATA | θ)
Bayes Method for Inference
Model for
DATA
DATA
Prior
f(θ)
Some Comments on Posterior Distributions
• The posterior f (θ |DATA) is function of the prior, the model,
and the data.
• In general, it is impossible to compute the multiple integral
R
L(DATA|θ )f (θ )dθ in closed form.
• New statistical and numerical methods that take advantage
of modern computing power are facilitating the computation of the posterior.
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Differences Between Bayesian and Frequentist
Inference
• Nuisance parameters
◮ Bayes methods use marginals.
◮ Large-sample likelihood theory suggest maximization.
• There are not important differences in large samples.
• Interpretation
◮ Bayes methods justified in terms of probabilities.
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◮ Frequentist methods justified on repeated sampling and
asymptotic theory.
Proper Prior Distributions
Any positive function defined on the parameter space that
integrates to a finite value (usually 1).
• Uniform prior: f (θ) = 1/(b − a) for a ≤ θ ≤ b.
This prior does not express strong preference for specific
values of θ in the interval.
• Examples of non-uniform prior distributions:
◮ Normal with mean at a and and standard deviation b.
◮ Beta between specified a and b with specified shape parameters (allows for a more general shape).
◮ Isosceles triangle with base (range between) a and b.
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For a positive parameter θ, may want to specify the prior
in terms of log(θ).
Effect of Using Vague (or Diffuse) Prior Distributions
• For a uniform prior f (θ) (possibly improper) across all possible values of θ
R(θ )f (θ )
R(θ )
f (θ |DATA) = R
=R
R(θ )f (θ )dθ
R(θ )dθ
which indicates that the posterior f (θ |DATA) is proportional
to the likelihood.
• The posterior is approximately proportional to the likelihood for a proper (finite range) uniform if the range is
large enough so that R(θ ) ≈ 0 where f (θ ) = 0.
• Other diffuse priors also result in a posterior that is approximately proportional to the likelihood if R(θ ) is large relative
to f (θ ).
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• Informative.
Sources of Prior Information
◮ Past data.
◮ Physical, chemical, and mechanical theory.
◮ Expert knowledge.
• Diffuse (or approximately non-informative).
◮ Uniform over finite range of parameter (or function of
parameter).
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◮ Uniform over infinite range of parameter (improper prior).
◮ Other vague or diffuse priors.
Improper Prior Distributions
f (θ)dθ = ∞,
Positive function f (θ) over parameter space for which
Z
• Uniform in an interval of infinite length: f (θ) = c for all θ.
• For a positive parameter θ the corresponding choice is
f [log(θ)] = c and f (θ) = (c/θ), θ > 0.
f (θ)L(θ|DATA)dθ < ∞
To use an improper prior, one must have
Z
(a condition on the form of the likelihood and the DATA).
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• These prior distributions can be made to be proper by specification of a finite interval for θ and choosing c such that
the total probability is 1.
Eliciting or Specifying a Prior Distribution
• The elicitation of a meaningful joint prior distribution for
vector parameters may be difficult
◮ The marginals may not completely determine the joint
distribution.
◮ Difficult to express/elicit dependences among parameters through a joint distribution.
◮ The standard parameterization may not have practical
meaning.
• General approach: choose an appropriate parameterization
in which the priors for the parameters are approximately
independent.
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With appropriate questioning, engineers provided the following information:
Expert Opinion and Eliciting Prior Information
• Identify parameters that, from past experience (or data),
can be specified approximately independently (e.g., for high
reliability applications a small quantile and the Weibull shape
parameter).
• Time to fracture data can often be described by a Weibull
distribution.
Example of Eliciting Prior Information: Bearing-Cage
Time to Fracture Distribution
• Determine for which parameters there is useful informative
prior information.
• From previous similar studies involving heavily censored data,
(µ, σ) tend to be correlated (making it difficult to specify a
joint prior for them).
p
σγ/2 × σ(1−γ/2) ,
b0 = log
i
σ(1−γ/2) /σγ/2 /z(1−γ/2)
hp
0
1000
3000
.01 quantile
5000
0.3
0.4
sigma [log axis]
0.5
• The prior for log(σ) is normal
"
0.20
0.30
log(σ) − a0
1
,
φnor
b0
b0
1
,
log(b1/a1)
• The prior for log(tp) is uniform
f [log(tp)] =
f (tp, σ) =
f [log(tp)] f [log(σ)]
tp
σ
sigma
0.40
σ > 0.
0.50
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a1 ≤ tp ≤ b1, σ > 0.
• Consequently, the joint prior distribution for (tp, σ) is
The corresponding density for tp is f (tp) = (1/tp)f [log(tp)].
a1 ≤ tp ≤ b 1 .
The corresponding density for σ is f (σ) = (1/σ)f [log(σ)].
f [log(σ)] =
#
Joint Lognormal-Uniform Prior Distributions
0.2
Prior pdfs for log(σ) and σ when σ0.005 = 0.2, σ0.995 = 0.5
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• For small p (near the proportion failing in previous studies), (tp, σ) are approximately independent (which allows for
specification of approximately independent priors).
• For parameters for which there in no useful informative prior
information, determine the form and range of the vague
prior (e.g., uniform over a wide interval).
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• For parameters for which there is useful informative prior
information, specify the form and range of the distribution
(e.g., lognormal with 99.7% content between two specified
points).
Example of Eliciting Prior Information: Bearing-Cage
Fracture Field Data (Continued)
• Based on experience with previous products of the same
material and knowledge of the failure mechanism, there is
strong prior information about the Weibull shape parameter.
• The engineers did not have strong prior information on possible values for the distribution quantiles.
a0 = log
• For the Weibull shape parameter log(σ) ∼ NOR(a0, b0),
where a0 and b0 are obtained from the specification of two
quantiles σγ/2 and σ(1−γ/2) of the prior distribution for σ.
Then
5000
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• Uncertainty in the Weibull 0.01 quantile will be described
by UNIFORM[log(a1), log(b1)] distribution where a1 = 100
and b1 = 5000 (wide range—not very informative).
.01 quantile [log axis]
500
Prior pdfs for log(t0.01) and t0.01 when
a1 = 100, b1 = 5000
100
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Joint Prior Distribution for (µ, σ)
−1
• The transformation µ = log(tp) − Φsev
(p)σ, σ = σ yields the
prior for (µ, σ)
f [log(tp)] f [log(σ)]
f (µ, σ) =
×
× tp
tp
σ
f [log(σ)]
= f [log(tp)] ×
σ
1
φnor {[log(σ) − a0] /b0}
=
×
log(b1/a1)
σb0
−1
−1
where log(a1) − Φsev
(p)σ ≤ µ ≤ log(b1) − Φsev
(p)σ, σ > 0.
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• The region in which f (µ, σ) > 0 is South-West to North−1
(p)Var(σ) > 0.
East oriented because Cov(µ, σ) = −Φsev
Methods to Compute the Posterior
• Numerical integration: to obtain the posterior, one needs
R
to evaluate the integral f (θ |DATA) =
R(θ )f (θ )dθ over
the region on which f (θ ) > 0.
In general there is not a closed form for the integral and the
computation has to done numerically using fixed quadrature
or adaptive integration algorithms.
0.50
0.45
0.40
0.35
0.30
0.25
0.20
Simulated Joint Prior for t0.01 and σ
•
200
1000
0.01 quantile
500
2000
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100
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Joint Posterior Distribution for (µ, σ)
i=1
δi 1−δi
2003 Y
1
log(ti ) − µ
log(ti ) − µ
× 1 − Φsev
φsev
σti
σ
σ
• The likelihood is
L(µ, σ) =
where δi indicates whether the observation i is a failure or
a right censored observation.
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R(µ, σ)f (µ, σ)
L(µ, σ)f (µ, σ)
=RR
.
L(v, w)f (v, w)dvdw
R(v, w)f (v, w)dvdw
• The posterior distribution is
f (µ, σ|DATA) = R R
Computing the Posterior Using Simulation
Using simulation, one can draw a sample from the posterior
using only the likelihood and the prior. The procedure for a
general parameter θ and prior distribution f (θ ) is as follows:
• Let θ i, i = 1, . . . , M be a random sample from f (θ ).
• The ith observation, θ i, is retained with probability R(θ i).
Ui ≤ R(θ i).
Then if Ui is a random observation from a uniform (0, 1),
θ i is retained if
0.5
0.3
0.1
5
6
mu
9
9
11
0.2
sigma
Simulated Joint and Marginal
Prior Distributions for µ and σ
7
mu
8
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0.4
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0.6
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⋆ , . . . θ⋆
• It can be shown that the retained observations, say θ 1
M⋆
(M ⋆ ≤ M ) are observations from the posterior f (θ |DATA).
sigma
• Simulation methods: the posterior can be approximated
using Monte Carlo simulation resampling methods.
sigma
Sampling from the Prior
The joint prior for θ = (µ, σ), is generated as follows:
i = 1, . . . , M
• Use the inverse cdf method (see Chapter 4) to obtain a
pseudorandom sample for tp, say
U
(tp)i = a1 × b1 1i ,
where U11, . . . , U1M are a pseudorandom sample from a uniform (0, 1).
i
• Similarly, obtain a pseudorandom sample for σ, say
h
−1 (U )
σi = exp a0 + b0Φnor
2i
where U21, . . . , U2M are another independent pseudorandom
sample from a uniform (0, 1).
7
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8
•
Joint Posterior for µ and σ
6
Z
f (θ )R(θ )dθ
•
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11
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0.5 0.1 0.01
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Simulated Joint Prior Distribution with µ and σ
Relative Likelihood
0.6
0.5
0.4
0.3
0.2
0.1
0.60
0.50
0.40
0.30
0.20
7.5
8.0
mu
9.5
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mu
9.0
•
10.5
10.0
Z
0.2
0.3
0.4
sigma
f (θ |DATA)dθ ′.
0.5
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• Inferences for µ or σ alone are based on the corresponding
marginal distributions µi⋆ and σi⋆, respectively.
• Using the general resampling method described above, one
gets a sample for the posterior for θ , say θ i⋆ = (µi⋆, σi⋆),
i = 1, . . . , M ⋆.
where θ ′ is the subset of the parameters excluding θj .
f [θ j |DATA] =
• Inferences on individual parameters are obtained by using
the marginal posterior distribution of the parameter of interest. The marginal posterior of θj is
Posterior and Marginal Posterior Distributions for the
Model Parameters
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Joint Posterior and Marginals for µ and σ for the
Bearing Cage Data
sigma
sigma
−1
• Then θ i = (µi, σi) with µi = log [(tp)i] − Φsev
(p)σi is a pseudorandom sample from the (µ, σ) prior.
0.6
0.5
0.4
0.3
0.2
0.1
5
E(M ⋆) = M
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• Can add to the posterior by sequentially filtering groups
of prior points until a sufficient number is available in the
posterior.
• When the prior and the data do not agree well, M ⋆ << M
otherwise and a larger prior sample will be required.
Consequently,
The number of observations M ⋆ from the posterior is random with an expected value of
Comments on Computing Posteriors Using Resampling
sigma
Posterior and Marginal Posterior Distributions for the
Functions of Model Parameters
• Inferences on a scalar function of the parameters g(θ ) are
obtained by using the marginal posterior distribution of the
functions of the parameters of interest, f [g(θ )|DATA].
• Using the simulation method, inferences are based on the
simulated posterior marginal distributions. For example:
◮ The marginal posterior distribution of f (tp|DATA) for
inference on quantiles is obtained from the empirical dis−1
tribution of µi⋆ + Φsev
(p)σi⋆.
◮ The marginal posterior distribution of f [F (te)|DATA] for
0.10
0.15
F(2000 | DATA)
0.20
0.0
0.2
0.4
0.6
F(5000 | DATA)
0.8
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1.0
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inference for failure probabilities at te is obtained
from
log(te )−µi⋆
.
σi⋆
the empirical distribution of Φsev
0.05
Simulated Marginal Posterior Distributions
for F (2000) and F (5000)
0.0
One-Sided Bayes Confidence Bounds
f [g(θ )|DATA]dg(θ ) = 1 − α
e
• A 100(1 − α)% Bayes lower confidence bound (or credible
bound) for a scalar function g(θ ) is value g satisfying
Z ∞
g
e
f [g(θ )|DATA]dg(θ ) = 1 − α
• A 100(1 − α)% Bayes upper confidence bound (or credible
bound) for a scalar function g(θ ) is value g̃ satisfying
Z g̃
−∞
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6000
.05 quantile (Hours)
4000
8000
2000
6000
.10 quantile (Hours)
4000
Simulated Marginal Posterior Distributions
for t0.05 and t0.10
2000
Bayes Point Estimation
8000
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Bayesian inference for θ and functions of the parameters
g (θ ) are entirely based on their posterior distributions f (θ |DATA)
and f [g (θ )|DATA].
Point Estimation:
Z
g(θ )f (θ |DATA)dθ .
• If g(θ ) is a scalar, a common Bayesian estimate of g(θ ) is
its posterior mean, which is given by
gb(θ ) = E[g(θ )|DATA] =
In particular, for the ith component of θ , θ̂i is the posterior
mean of θi. This estimate is the the Bayes estimate that
minimizes the square error loss.
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• Other possible choices to estimate g(θ ) include (a) the posterior mode, which is very similar to the ML estimate and
(b) the posterior median.
Two-Sided Bayes Confidence Intervals
f [g(θ )|DATA]dg(θ ) = 1 − α
e
(1)
• A 100(1 − α)% Bayes confidence interval (or credible interval) for a scalar function g(θ ) is any interval [g , g̃] satisfying
g
Z g̃
e
• The interval [g , e g̃] can be chosen in different ways
◮ Combining two 100(1 − α/2)% intervals puts equal probability in each tail (preferable when there is more concern
for being incorrect in one direction than the other).
e
◮ A 100(1 − α)% Highest Posterior Density (HPD) confidence interval chooses [g , g̃] to consist of all values of
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g with f (g|DATA) > c where c is chosen such that (1)
holds. HPD intervals are similar to likelihood-based confidence intervals. Also, when f [g(θ )|DATA] is unimodal
the HPD is the narrowest Bayes interval.
Bayesian Joint Confidence Regions
The same procedure generalizes to confidence regions for
vector functions g (θ ) of θ .
• A 100(1 − α)% Bayes confidence region (or credible region)
for a vector valued function g (θ ) is defined as
CRB = {g (θ )|f [g|DATA] ≥ c}
f [g (θ )|DATA)dg (θ ) = 1 − α
where c is chosen such that
Z
CRB
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• In this case the presentation of the confidence region is
difficult when θ has more than 2 components.
Prediction of Future Events
• Future events can be predicted by using the Bayes predictive
distribution.
• If X [with pdf f (·|θ )] represents a future random variable
f (x|θ )f (θ |DATA)dθ
◮ the posterior predictive pdf of X is
Z
f (u|θ )du =
Z
F (x|θ )f (θ |DATA)dθ
= Eθ|DATA [f (x|θ )]
f (x|DATA) =
−∞
Z x
◮ the posterior predictive cdf of X is
F (x|DATA) =
= Eθ|DATA [F (x|θ )]
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where the expectations are computed with respect to the
posterior distribution of θ .
Location-Scale Based Prediction Problems
Here we consider prediction problems when log(T ) has a
location-scale distribution.
F (t|θ ) = Φ(ζ)
• Predicting a future value of T . In this case, X = T and
x = t, then
1
φ(ζ),
f (t|θ ) =
σt
where ζ = [log(t) − µ]/σ.
⋆
X 1
1 M
φsev (ζi⋆)
M ⋆ i=1 σi⋆t
⋆
X
1 M
Φsev (ζi⋆)
M ⋆ i=1
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• Thus, for the Bearing-cage fracture data, approximations
of the predictive pdf and cdf for a new observation are:
f (t|DATA) ≈
F (t|DATA) ≈
where ζi⋆ = [log(t) − µi⋆]/σi⋆.
Bayes Versus Likelihood
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• Summary table or plots to compare the Likelihood versus
the Bayes Methods to compare confidence intervals for µ,
σ, and t0.1 for the Bearing-cage data example.
Approximating Predictive Distributions
X
1 M
f (x|θ i⋆).
M ⋆ i=1
⋆
• f (x|DATA) can be approximated by the average of the posterior pdfs f (x|θ i⋆). Then
f (x|DATA) ≈
X
1 M
F (x|θ i⋆).
M ⋆ i=1
⋆
• Similarly, F (x|DATA) can be approximated by the average
of the posterior cdfs F (x|θ i⋆). Then
F (x|DATA) ≈
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• A two-sided 100(1 − α)% Bayesian prediction interval for a
new observation is given by the α/2 and (1 − α/2) quantiles
of F (x|DATA).
Prediction of an Order Statistic
Here we consider prediction of the kth order statistic in a
future sample of size m from the distribution of T when
log(T ) has a location-scale distribution.
j=k
m
X
× [1 − Φ (ζ)]m−k
m!
[Φ (ζ)]j × [1 − Φ (ζ)]m−j
j! (m − j)!
1
m!
× [Φ (ζ)]k−1 ×
φ (ζ)
(k − 1)! (m − k)!
σt(k)
• In this case, X = T(k) and x = t(k), then
f [t(k)|θ ] =
F [t(k)|θ ] =
where ζ = [log(t(k)) − µ]/σ.
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Predicting the 1st Order Statistic
1
φ (ζ) × [1 − Φ (ζ)]m−1
σt(1)
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When k = 1 (predicting the 1st order statistic), the formulas
simplify to
• Predictive pdf
f [t(1)|θ ] = m ×
• Predictive cdf
F [t(1)|θ ] = 1 − [1 − Φ (ζ)]m
where ζ = [log(t(1)) − µ]/σ.
Predicting a New Observation
• F (t|DATA) can be approximated by the average of the posterior probabilities F (t|θ i⋆), i = 1, . . . , M ⋆.
• Similarly, f (t|DATA) can be approximated by the average
of the posterior densities f (t|θ i⋆), i = 1, . . . , M ⋆.
⋆
"
#
#
• In particular for the Bearing-cage fracture data, an approximation for the predictive pdf and cdf are
"
X
log(t) − µi⋆
1 M
Φsev
.
M ⋆ i=1
σi⋆
⋆
X 1
log(t) − µi⋆
1 M
f (t|DATA) ≈
φsev
M ⋆ i=1 σi⋆t
σi⋆
F (t|DATA) ≈
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• A 100(1 − α)% Bayesian prediction interval for a new observation is given by the percentiles of this distribution.
Caution on the Use of Prior Information
• In many applications, engineers really have useful, indisputable prior information. In such cases, the information
should be integrated into the analysis.
• We must beware of the use of wishful thinking as prior information. The potential for generating seriously misleading
conclusions is high.
• As with other inferential methods, when using Bayesian
methods, it is important to do sensitivity analyses with
respect to uncertain inputs to ones model (including the
inputted prior information)
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Predicting the 1st Order Statistic for the
Bearing-Cage Fracture Data
For the Bearing-cage fracture data:
(
)
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X
1 M
1
m × ⋆ φ ζi⋆ × 1 − Φ ζi⋆ m−1
M ⋆ i=1
σi t
⋆
• An approximation for the predictive pdf for the 1st order
statistic is
f [t(1)|DATA] ≈
⋆
• The corresponding predictive cdf is
X n
o
1 M
1 − 1 − Φ ζi⋆ m
F [t(k)|DATA] ≈
M ⋆ i=1
where ζi⋆ = [log(t) − µi⋆]/σi⋆.
5000
Hours
10000
15000
Predictive Density and Prediction Intervals for a
Future Observation from the Bearing Cage Population
0
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